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Properties

Label 2-690-115.114-c2-0-29
Degree $2$
Conductor $690$
Sign $0.985 - 0.167i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (1.15 + 4.86i)5-s + 2.44·6-s − 1.79·7-s − 2.82i·8-s − 2.99·9-s + (−6.87 + 1.63i)10-s − 0.382i·11-s + 3.46i·12-s − 24.0i·13-s − 2.53i·14-s + (8.42 − 2.00i)15-s + 4.00·16-s + 22.1·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (0.231 + 0.972i)5-s + 0.408·6-s − 0.256·7-s − 0.353i·8-s − 0.333·9-s + (−0.687 + 0.163i)10-s − 0.0347i·11-s + 0.288i·12-s − 1.84i·13-s − 0.181i·14-s + (0.561 − 0.133i)15-s + 0.250·16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.691612799\)
\(L(\frac12)\) \(\approx\) \(1.691612799\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + 1.73iT \)
5 \( 1 + (-1.15 - 4.86i)T \)
23 \( 1 + (1.50 - 22.9i)T \)
good7 \( 1 + 1.79T + 49T^{2} \)
11 \( 1 + 0.382iT - 121T^{2} \)
13 \( 1 + 24.0iT - 169T^{2} \)
17 \( 1 - 22.1T + 289T^{2} \)
19 \( 1 + 17.6iT - 361T^{2} \)
29 \( 1 + 1.79T + 841T^{2} \)
31 \( 1 - 40.3T + 961T^{2} \)
37 \( 1 - 42.7T + 1.36e3T^{2} \)
41 \( 1 - 1.26T + 1.68e3T^{2} \)
43 \( 1 - 16.5T + 1.84e3T^{2} \)
47 \( 1 + 5.75iT - 2.20e3T^{2} \)
53 \( 1 - 81.4T + 2.80e3T^{2} \)
59 \( 1 - 18.0T + 3.48e3T^{2} \)
61 \( 1 + 33.1iT - 3.72e3T^{2} \)
67 \( 1 - 116.T + 4.48e3T^{2} \)
71 \( 1 - 28.3T + 5.04e3T^{2} \)
73 \( 1 + 112. iT - 5.32e3T^{2} \)
79 \( 1 - 62.0iT - 6.24e3T^{2} \)
83 \( 1 - 148.T + 6.88e3T^{2} \)
89 \( 1 + 102. iT - 7.92e3T^{2} \)
97 \( 1 + 150.T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15207790018136578733095557340, −9.510768968892252311569463045750, −8.118475857489181693372553381703, −7.68495250566484549825342866217, −6.77836791815908715637339349391, −5.92572572013579456717647039106, −5.23994823266585115797624977439, −3.51348649842950132801830751785, −2.69254990069190433771873084197, −0.77518144416926945420857911922, 1.06378332328823057313496624105, 2.36937551006125553173938449832, 3.84830938254432667223754642269, 4.48481352274715240555185186362, 5.50004144702566336375725843280, 6.51024426837785326364637798970, 8.013237531106281645950124704671, 8.735275408138668704832993684791, 9.691845251109655683714713890732, 9.921550280419170322568396450306

Graph of the $Z$-function along the critical line